Object does cast shadow, but doesn't get physically darker
I'm not sure what's going on. I downloaded your files, I opened the file, I had to go to External Data - Find Missing Files, but that's just because I moved them, I think. Either way, without changing anything else, this is what I get:

So, I'm not sure if there's something wrong with your version of blender, or graphics drivers or whatnot, but it seems fine on my end.  For testing purposes, as well as easier shareability, I am including the .blend file in this post with the textures packed.
File is here - 
If you're still having problems, one thing I can recommend is to swap the sun lamp for a large area light - I always find it casts better shadows and doesn't give such uniform brightness, but that's just me.
Hope this helps.
EDIT - Because you mentioned the normal being a problem, I looked closer at it and noticed your normal map is connected through a Normal node, as opposed to a Normal Map node as it should be.  I have included an image below of the correct node setup, for reference.